Actual Infinity in Reality?

[GrandizerII]

Steve seems to have disappeared from the thread, either because he has realized he was wrong, or because he still thinks he’s right.  Or, I dunno; maybe he needed to do laundry or something.  😛

Actually sailing on a 46' sailboat for a week as part of sailing certification.

So you left us all hanging here in this thread for a while just to get your bloody certification in sailing?

[polymath257]

The *best* answer to the question 'what is the universe expanding into?' is THE FUTURE.

The universe is expanding into the future.

And, while that may sound like a trick of language, there is a strict sense in which it is completely correct in general relativity.

So, as discussed above, spacetime is a four dimensional construct: it consists of all of space and all of time as a single geometric entity. Spatial cross sections of this entity (i.e, space) are 'larger' for later times than they are for earlier times. That is what is means to say space is expanding. It just means that a later 3 dimensional cross section is larger (in some sense) that an earlier one.

A very basic analogy may make this easier to comprehend. Suppose that the surface of the earth represents spacetime. In this analogy, think of higher latitudes as being later in time. Different longitudes are different spatial locations.

So, a 'time slice' is a cross section of the surface of the earth that has the same latitude: a latitude line. This represents space at a specific time.
 

Now, in the southern hemisphere, as we move north (which is later in time), the latitude lines get larger: space is expanding. In the northern hemisphere, as we move north, the latitude lines get smaller: space is contracting.

Also, we have a 'beginning' at the south pole and an 'end' at the north pole. In this analogy, time is finite with 'singularities' at the poles.

Now, what are the latitude lines expanding into? Different latitude lines aren't even in the 'same space' because they are at different times. They expand or contract into the future!

Now, this is a *very* simplistic analogy, but the essences are there: spacetime is a single geometric construct, space corresponds to cross sections of that geometry. Different times correspond to different cross sections, and expansion/contraction are determined by comparing two different cross sections.

But then what about the earth itself and beyond? There is nevertheless still something beyond "finite space" in this analogy (which, as you say, is a simplistic one anyway). So even if we were like 2D beings who couldn't understand the notion of any sort of direction in the third dimension or whatever, and weren't aware that the "latitude timeslices" were placed in the context of a planet that is situated in an outer space, it does seem like even in our ignorance of what it is exactly, we still have to logically conclude there has to be something out there, as opposed to literally nothing.

Perhaps that's not what you were saying, but I'm just saying it here just to let you know where I'm coming from here.

In this analogy, we would be *one* dimensional beings.

You are, essentially, assuming that spacetime has to be embedded into something larger. But in this analogy, spacetime *is* just the surface of the Earth.

Perhaps a different analogy.

In this analogy, we use three dimensional space as our analogy for spacetime, but in this, the *radial* dimension from some central point is our time variable. So, later times are those farther away from the center.

What does a time cross section now look like? It is the *surface* of all points some fixed distance from the center: a sphere.

For later times (again, radius), the sphere is larger. So, space is 'expanding' into the future. Here, *space* is a sphere (not all of 3D, which is spacetime).

In this analogy, we (at a single time, that is) would be two dimensional beings on the surface of a sphere.

Now, this model has a 'beginning' (at the center) with no time prior to that. It is infinite into the future and space is finite at all times, but expanding.

One amusing aspect of this analogy is that spacetime is 'flat' while space is curved. In the previous analogy, both were curved. In cosmology, we talk about a flat space and curved spacetime.

[Jehanne]

The Universe is growing in spatial extent, since, on an intergalactic scale, distances are increasing.  Even if the Universe is infinite in spatial extent, the distances between any two galaxies is finite, and so, one has Cantor's concept of "infinities within infinities":

https://blogs.scientificamerican.com/the...thematics/

A Universe that is an "actual infinite" in space and time is completely reasonable.

Yes to this.

Yes to a multiverse.

No to a finite universe expanding into nothing.

Glad you brought this up though because all the more reason to go with actual infinity rather than against.

Dr. Craig (and, Steve) are falling into the Fallacy of Decomposition -- Just because every member of a flock of sheep has a mother, it does not follow that "the flock" has "a mother".

[LadyForCamus]

Steve seems to have disappeared from the thread, either because he has realized he was wrong, or because he still thinks he’s right.  Or, I dunno; maybe he needed to do laundry or something.  😛

Actually sailing on a 46' sailboat for a week as part of sailing certification.

That sounds way cooler than doing laundry!  😁

[GrandizerII]

But then what about the earth itself and beyond? There is nevertheless still something beyond "finite space" in this analogy (which, as you say, is a simplistic one anyway). So even if we were like 2D beings who couldn't understand the notion of any sort of direction in the third dimension or whatever, and weren't aware that the "latitude timeslices" were placed in the context of a planet that is situated in an outer space, it does seem like even in our ignorance of what it is exactly, we still have to logically conclude there has to be something out there, as opposed to literally nothing.

Perhaps that's not what you were saying, but I'm just saying it here just to let you know where I'm coming from here.

In this analogy, we would be *one* dimensional beings.

You are, essentially, assuming that spacetime has to be embedded into something larger. But in this analogy, spacetime *is* just the surface of the Earth.

Perhaps a different analogy.

In this analogy, we use three dimensional space as our analogy for spacetime, but in this, the *radial* dimension from some central point is our time variable. So, later times are those farther away from the center.

What does a time cross section now look like? It is the *surface* of all points some fixed distance from the center: a sphere.

For later times (again, radius), the sphere is larger. So, space is 'expanding' into the future. Here, *space* is a sphere (not all of 3D, which is spacetime).

In this analogy, we (at a single time, that is) would be two dimensional beings on the surface of a sphere.

Now, this model has a 'beginning' (at the center) with no time prior to that. It is infinite into the future and space is finite at all times, but expanding.

One amusing aspect of this analogy is that spacetime is 'flat' while space is curved. In the previous analogy, both were curved. In cosmology, we talk about a flat space and curved spacetime.

Yeah, I was doing some reading on this some few hours ago and came upon the balloon analogy to explain how a finite universe could expand into "nothing". The flat universe seems to be almost a given now in astrophysics, am I right? I find this one really interesting as I always thought of it as a sphere ...

Anyhow, the analogy is good at illustrating what it could be like for a finite universe to expand "out of nothing" and/or "into nothing", but then in this case, the way I avoid the problematic existence of "non-existence" beyond spacetime is to posit a wider cosmos (perhaps a multiverse of universes), one from which finite universes spring forth into existence. Perhaps each time slice is like a frame in a 4D (or higher) kind of reality, and so that reality is still something into which the 3D sphere is embedded. Finite totality of existence seems to imply the existence of "non-existence" beyond it, and I find that problematic. Anything about which you can say is beyond or outside this or that is a something as far as I'm concerned, even if it's not the same nature as the spacetime we speak of here.

[polymath257]

In this analogy, we would be *one* dimensional beings.

You are, essentially, assuming that spacetime has to be embedded into something larger. But in this analogy, spacetime *is* just the surface of the Earth.

Perhaps a different analogy.

In this analogy, we use three dimensional space as our analogy for spacetime, but in this, the *radial* dimension from some central point is our time variable. So, later times are those farther away from the center.

What does a time cross section now look like? It is the *surface* of all points some fixed distance from the center: a sphere.

For later times (again, radius), the sphere is larger. So, space is 'expanding' into the future. Here, *space* is a sphere (not all of 3D, which is spacetime).

In this analogy, we (at a single time, that is) would be two dimensional beings on the surface of a sphere.

Now, this model has a 'beginning' (at the center) with no time prior to that. It is infinite into the future and space is finite at all times, but expanding.

One amusing aspect of this analogy is that spacetime is 'flat' while space is curved. In the previous analogy, both were curved. In cosmology, we talk about a flat space and curved spacetime.

Yeah, I was doing some reading on this some few hours ago and came upon the balloon analogy to explain how a finite universe could expand into "nothing". The flat universe seems to be almost a given now in astrophysics, am I right? I find this one really interesting as I always thought of it as a sphere ...

Anyhow, the analogy is good at illustrating what it could be like for a finite universe to expand "out of nothing" and/or "into nothing", but then in this case, the way I avoid the problematic existence of "non-existence" beyond spacetime is to posit a wider cosmos (perhaps a multiverse of universes), one from which finite universes spring forth into existence. Perhaps each time slice is like a frame in a 4D (or higher) kind of reality, and so that reality is still something into which the 3D sphere is embedded. Finite totality of existence seems to imply the existence of "non-existence" beyond it, and I find that problematic. Anything about which you can say is beyond or outside this or that is a something as far as I'm concerned, even if it's not the same nature as the spacetime we speak of here.

Which ultimately means that the concept of 'spacetime' as literally *all* of space and *all* of time as a single entity is part of your difficulty. Absent a multiverse (which only pushes the problem up one level since in almost every multiverse model, the multiverse is expanding), everything is part of spacetime.

There is no 'outside'. And expansion is just to the next cross section, not into anything 'outside'.

[GrandizerII]

Yeah, I was doing some reading on this some few hours ago and came upon the balloon analogy to explain how a finite universe could expand into "nothing". The flat universe seems to be almost a given now in astrophysics, am I right? I find this one really interesting as I always thought of it as a sphere ...

Anyhow, the analogy is good at illustrating what it could be like for a finite universe to expand "out of nothing" and/or "into nothing", but then in this case, the way I avoid the problematic existence of "non-existence" beyond spacetime is to posit a wider cosmos (perhaps a multiverse of universes), one from which finite universes spring forth into existence. Perhaps each time slice is like a frame in a 4D (or higher) kind of reality, and so that reality is still something into which the 3D sphere is embedded. Finite totality of existence seems to imply the existence of "non-existence" beyond it, and I find that problematic. Anything about which you can say is beyond or outside this or that is a something as far as I'm concerned, even if it's not the same nature as the spacetime we speak of here.

Which ultimately means that the concept of 'spacetime' as literally *all* of space and *all* of time as a single entity is part of your difficulty. Absent a multiverse (which only pushes the problem up one level since in almost every multiverse model, the multiverse is expanding), everything is part of spacetime.

There is no 'outside'. And expansion is just to the next cross section, not into anything 'outside'.

If it's an infinite universe expanding, there's no "outside" problem. If the totality of existence is finite, however, then yes, there is an "outside" that is "nonexistence", even if the expansion itself is just to the next cross section.

Just trying to counter this with words like "there's no outside, everything that exists is all contained in the spacetime" doesn't negate the logical mandate that there is an "outside". If we go with the premise, we have to follow through with it to the conclusion implied. We can't just stop at the premise and be satisfied.

[polymath257]

Which ultimately means that the concept of 'spacetime' as literally *all* of space and *all* of time as a single entity is part of your difficulty. Absent a multiverse (which only pushes the problem up one level since in almost every multiverse model, the multiverse is expanding), everything is part of spacetime.

There is no 'outside'. And expansion is just to the next cross section, not into anything 'outside'.

If it's an infinite universe expanding, there's no "outside" problem. If the totality of existence is finite, however, then yes, there is an "outside" that is "nonexistence", even if the expansion itself is just to the next cross section.

Just trying to counter this with words like "there's no outside, everything that exists is all contained in the spacetime" doesn't negate the logical mandate that there is an "outside". If we go with the premise, we have to follow through with it to the conclusion implied. We can't just stop at the premise and be satisfied.

OK, so to be clear. There is no 'outside' of spacetime. There is an outside of any particular spatial cross section (any other spatial cross section). But the collection of *all* such cross sections is 'spacetime'.

And again, the best answer to what the universe is expanding into, even for finite space, is 'the future'.

---

Hmmm...a finite space does NOT imply an outside. That may be one of the issues here.

So, suppose that space is curved. In practice, what that means is that whatever direction you set off, if you keep going you will eventually come back around to your starting point (no actual travel--stay on one spatial cross section). So, you decide to take off in the 'up' direction and travel for a few billion light years, you will come back around to the start from the 'south' direction. Same with East vs West, and any direction and its opposite.

The reason I say not to actually travel is the time aspect of such: motion requires time and the time it would take to go 'around' is enough that the size changes during the trip.

So, just take a specific spatial cross section and go off in one direction, staying on that cross section. Eventually, with a finite space, you will come back around.

---

Also, don't confuse the 'totality of all existence' with the 'totality of space at this time'.

[GrandizerII]

If it's an infinite universe expanding, there's no "outside" problem. If the totality of existence is finite, however, then yes, there is an "outside" that is "nonexistence", even if the expansion itself is just to the next cross section.

Just trying to counter this with words like "there's no outside, everything that exists is all contained in the spacetime" doesn't negate the logical mandate that there is an "outside". If we go with the premise, we have to follow through with it to the conclusion implied. We can't just stop at the premise and be satisfied.

OK, so to be clear. There is no 'outside' of spacetime. There is an outside of any particular spatial cross section (any other spatial cross section). But the collection of *all* such cross sections is 'spacetime'.

And again, the best answer to what the universe is expanding into, even for finite space, is 'the future'.

---

Hmmm...a finite space does NOT imply an outside. That may be one of the issues here.

So, suppose that space is curved. In practice, what that means is that whatever direction you set off, if you keep going you will eventually come back around to your starting point (no actual travel--stay on one spatial cross section). So, you decide to take off in the 'up' direction and travel for a few billion light years, you will come back around to the start from the 'south' direction. Same with East vs West, and any direction and its opposite.

The reason I say not to actually travel is the time aspect of such: motion requires time and the time it would take to go 'around' is enough that the size changes during the trip.

So, just take a specific spatial cross section and go off in one direction, staying on that cross section. Eventually, with a finite space, you will come back around.

---

Also, don't confuse the 'totality of all existence' with the 'totality of space at this time'.

I'm still not seeing it, but since it's late at night here, let me think about it in the morning, and do some extra reading. And pretty much every physicist seems to agree with you as well ... ugh ...

[GrandizerII]

If it's an infinite universe expanding, there's no "outside" problem. If the totality of existence is finite, however, then yes, there is an "outside" that is "nonexistence", even if the expansion itself is just to the next cross section.

Just trying to counter this with words like "there's no outside, everything that exists is all contained in the spacetime" doesn't negate the logical mandate that there is an "outside". If we go with the premise, we have to follow through with it to the conclusion implied. We can't just stop at the premise and be satisfied.

OK, so to be clear. There is no 'outside' of spacetime. There is an outside of any particular spatial cross section (any other spatial cross section). But the collection of *all* such cross sections is 'spacetime'.

And again, the best answer to what the universe is expanding into, even for finite space, is 'the future'.

---

Hmmm...a finite space does NOT imply an outside. That may be one of the issues here.

So, suppose that space is curved. In practice, what that means is that whatever direction you set off, if you keep going you will eventually come back around to your starting point (no actual travel--stay on one spatial cross section). So, you decide to take off in the 'up' direction and travel for a few billion light years, you will come back around to the start from the 'south' direction. Same with East vs West, and any direction and its opposite.

The reason I say not to actually travel is the time aspect of such: motion requires time and the time it would take to go 'around' is enough that the size changes during the trip.

So, just take a specific spatial cross section and go off in one direction, staying on that cross section. Eventually, with a finite space, you will come back around.

---

Also, don't confuse the 'totality of all existence' with the 'totality of space at this time'.

Ok, so what you're saying (in the case of a finite universe) is that 'the totality of space at a certain time' is finite in the sense that eventually if you "movelessly" go across its full length (or whatever) in one direction, you eventually reach the point where you first began "movelessly" going (so basically a loop). Sort of like the surface of a ball, but we need to think in a higher dimension instead. IOW, finite does not always imply "edge". Did I get you right?